Gorensteinprojective modules over nakayama algebras deformations of gorensteinprojective modules over a nakayama algebra and triangular matrix algebra j. A brief introduction to gorenstein projective modules 7 denote by sgproj the full subcategory of all the modules m isomorphic to kerf, where. Ris closed under arbitrary direct sums and under direct summands. The gorensteinprojective modules over a monomial algebra. Pdf ding projective and ding injective modules researchgate. We mention that we rely on a fundamental result in 18, which implies in particular. We prove that over an n gorenstein ring every complex has a gorenstein injective envelope and we show that every such envelope is a quasiisomorphim when the ring is commutative, local and gorenstein, auslander announced that every finitely generated r.
X,ygorenstein projective and injective modules academic journals. There are many equivalent conditions, some of them listed below, often saying that a gorenstein ring is selfdual in some sense. Chen, coherent ring with finite self fpinjective dimension, comm. Throughout the paper, all rings are associative with identity, and an module will mean right module.
We also investigate some connections between gorenstein projective, injective and. In basic homological algebra, projective, injective and flat modules play an important and fundamental role. Just as the gdimension has strong similarities with the projective dimension, the gorenstein injective and. Mahdou 5 studied a particular case of gorenstein projective, injective and. Frobenius categories, gorenstein algebras and rational surface singularities osamu iyama, martin kalck, michael wemyss, and dong yang dedicated to ragnarolaf buchweitz on the occasion of his 60th birthday. The gorensteinprojective modules in this section, we recall some basic facts on gorensteinprojective modules over a nitedimensional algebra. Jenda department of mathematicsaca, auburn university, al, 368495307.
On gorenstein projective and gorenstein flat modules. It was shown there that the class of gorenstein acprojective modules form the co. Pdf n strongly gorenstein projective, injective and. It turns out that there is an explicit classi cation of indecomposable gorenstein projective a modules, so that we unify the results in 17 and 15 to some extent. Strongly gorenstein projective, injective, and flat. The weak gorenstein injective and flat modules are defined. We introduce and study the gorenstein projective modules, where is a projective class. In classical homological algebra, the projective, injective and flat dimensions of modules are important and fundamental. A module m is gorenstein projective, if there exists a complete projective resolution p. A module is said to be gorenstein injective gorenstein projective if it is a cycle in a totally acyclic complex of injective projective modules. Gorenstein injective, projective and flat precovers. Gorenstein projective modules for the working algebraist. Auslander shows that over a commutative gorenstein local ring r, a finitely generated module m is gorenstein projective if and only if. Pdf an rmodule m is called ding projective if there exists an exact.
Let us to denote the class of rmodules with at dimension at most a xed nonnegative integer nby f n. Then they proved that an nstrongly gorenstein projective module. Gorenstein rings were introduced by grothendieck in his 1961 seminar published in hartshorne 1967. We extend this result to the category of complexes. Pureinjectivity in the category of gorenstein projective. Gorenstein projective modules enochs and jenda 1995 let r be a ring. The gorenstein semihereditary domains are called gorenstein prufer domains in 28. On gorenstein projective, injective and flat dimensionsa functorial. Browse other questions tagged commutativealgebra modules projectivemodule gorenstein or ask your own question. Cgorenstein projective, lcgorenstein injective and. Every flat module is a direct summand of a strongly gorenstein flat module. Thus this approach recovers the two standard model category structures the projective and injective on chain complexes over k, as well. We prove that the category of weak pureinjective gorenstein projective left rmodules coincides with the intersection of the category of pureinjective left rmodules and that of gorenstein projective left rmodules. Xiuhua luo ntu gorenstein projective modules april 26, 2018 3 16.
The class of all gorenstein projective r modules is denoted by gp. In this paper, weak gorenstein projective, injective and flat modules are introduced and investigated. We note that for a commutative, local gorenstein ring, the finitely generated gorenstein projective modules are just the maximal cohenmacaulay modules, so our. Strongly gorenstein projective, injective, and flat modules. In this paper, we study a particular case of gorenstein projective, injective, and flat modules, which we call, respectively, strongly gorenstein projective, injective, and flat modules.
Then, we get an equivalent characterization of virtually. Gorenstein injective and projective modules and actions of. Pdf a generalization of strongly gorenstein projective modules. Moreover, with such modules, we are able to give nice new characterizations of gorenstein projective, injective, and flat modules, similar to the characterization of projective modules by the free modules, which is the main result of this paper see theorem 2. T,n 0 for all m 0 and all rmodules t of finite projective or finite injective dimension. Gorenstein projective module, gorenstein injective module. We prove that over an ngorenstein ring every complex has a gorenstein injective envelope and we show that every such envelope is a quasiisomorphim when the ring is commutative, local and gorenstein, auslander announced that every finitely generated r. Applications of ngorenstein projective and injective modules xi tang y abstract. Enochs and his coauthors introduced gorenstein projective and injective modules and developed relative homological algebra. These modules are a generalization of the gorenstein projective modules.
We study the stability of gorenstein preenvelopes and precovers in the cases of hextensions and smash products with h,whereh is a hopf algebra. A module m is said to be strongly gorenstein at, if there exists a complete at resolution of the form f. In commutative algebra, a gorenstein local ring is a commutative noetherian local ring r with finite injective dimension as an rmodule. In this paper, we discuss some properties of gorenstein projective, injective and flat modules and study some connections between gorenstein injective and gorenstein flat modules. Applications of ngorenstein projective and injective modules. A left module q over the ring r is injective if it satisfies one and therefore all of the following equivalent conditions. In this paper, we introduce and study weak pureinjective gorenstein projective modules.
In this paper, we study a particular case of gorenstein projective, injective, and flat modules, which we call, respectively, strongly gorenstein projective, injective, 26 jan 2015 keywords. Over noetherian rings the gorenstein projective, injective, and flat modules were and. The gorensteinprojective modules over a monomial algebra 1117 tive amodules. Xgorenstein projective modules 491 now, assume that every rmodule is xgorenstein projective.
The standard reference on the representation theory of nitedimensional algebras is 3. Yang and liu discussed in 5 some connections among strongly gorenstein projective,injective and flat. If q is a submodule of some other left rmodule m, then there exists another submodule k of m such that m is the internal direct sum of q and k, i. This is the case when the ring is commutative noetherian with a dualizing complex. For two left amodules x and y, we denote by hom ax,y the space of mod ule homomorphisms from x to y, and by px,y the subspace formed by those homomorphisms factoring through projective modules. Pdf weak gorenstein projective, injective and flat modules. A left rmodule m is called weak gorenstein projective if there exists an exact sequence.
Gorenstein injective and projective modules springerlink. Moreover, we get that a virtually gorenstein artin algebra ris cm nite if and only if the category of pureinjective gorenstein projective left rmodules coincides with that of gorenstein projective left rmodules theorem 3. In this article we extend the notion of gorenstein injective and projective modules to that of complexes and characterize such complexes. We shall identify right amodules as left aopmodules, where aop is the opposite algebra of a. A short exact sequence of amodules is a sequence of the form 0. Foxby duality and gorenstein injective and projective modules.
Let p1 p0 m 0 and 0 m i0 i1 be projective and injective resolutions of m. Model structures on modules over dingchen rings gillespie, james, homology, homotopy and applications, 2010. Proper gorenstein projective resolutions a basic problem in gorenstein homological algebra is, given a ring r, when rgproj is convariantly. Let gpr be the category of gorenstein projective modules. These last three classes of modules give us a new characterization of the. We are interested in the gorensteinprojective modules over a monomial algebra a. The methods of gorenstein homological algebra play a part in commutative and. The study of gorenstein dimensions takes cues from the classical situation. K 0 any short exact sequence 0 q m k 0 of left rmodules splits. The principal role of the strongly gorenstein projective modules is to give a simple characterization of gorenstein projective modules. A left r module m is called weak gorenstein projective if there exists an exact sequence. The dual result for injective and gorenstein injective modules also holds over any ring r 2, proposition 4. Moreover, the authors consider the tcrprojective and lcr.